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Asymptotic expansion for harmonic functions in the half-space with a pressurized cavity. (English) Zbl 1350.35058
Summary: In this paper, we address a simplified version of a problem arising from volcanology. Specifically, as a reduced form of the boundary value problem for the Lamé system, we consider a Neumann problem for harmonic functions in the half-space with a cavity \(C\). Zero normal derivative is assumed at the boundary of the half-space; differently, at \(\partial C\), the normal derivative of the function is required to be given by an external datum \(g\), corresponding to a pressure term exerted on the medium at \(\partial C\). Under the assumption that the (pressurized) cavity is small with respect to the distance from the boundary of the half-space, we establish an asymptotic formula for the solution of the problem. Main ingredients are integral equation formulations of the harmonic solution of the Neumann problem and a spectral analysis of the integral operators involved in the problem. In the special case of a datum \(g\), which describes a constant pressure at \(\partial C\), we recover a simplified representation based on a polarization tensor.

35C20 Asymptotic expansions of solutions to PDEs
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
35J25 Boundary value problems for second-order elliptic equations
35Q86 PDEs in connection with geophysics
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